Interpolation Conditions for Instant Data Consistency with Port-Hamiltonian Structure

Martina Vanelli; Nima Monshizadeh; Julien M. Hendrickx

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Interpolation Conditions for Instant Data Consistency with Port-Hamiltonian Structure

Martina Vanelli, Nima Monshizadeh, Julien M. Hendrickx

arXiv:2608.31092v1Aug 31, 2026math.OCeess.SY

TL;DR

The paper addresses how to determine whether nonlinear measured data are consistent with physically structured port-Hamiltonian dynamics without explicit parametrization. It uses interpolation conditions to derive necessary and sufficient data-only tests and an SDP verification formulation for full-rank structures. The framework establishes a basis for future data-driven passivity-based control design, while remaining pointwise rather than trajectory-coherent.

Problem
Nonlinear data-driven methods often use restrictive system classes or parametrized representations, motivating consistency tests that combine measurements with physical structure without explicit parametrization.
Method
The paper applies interpolation conditions for smooth and convex Hamiltonian classes to characterize instant pH consistency and reformulates the full-rank verification problem as a convex semidefinite program.
Results
The framework provides necessary and sufficient algebraic conditions for instant consistency and a convex semidefinite programming formulation for efficient full-rank verification from data.
Takeaways & Limitations
The framework provides a principled way to assess physical consistency directly from measured trajectories and supports future data-driven passivity-based control design.
Takeaways & Limitations
The current framework is pointwise and does not enforce trajectory-level coherence; extending it to noisy and trajectory-consistent settings remains future work.

Abstract

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We develop a data-driven framework for nonlinear port-Hamiltonian (pH) systems based on interpolation conditions to characterize consistency between observed data and structured dynamical models. Specifically, we derive necessary and sufficient conditions for the existence of a pH system with a smooth (convex) Hamiltonian instantly consistent with a given dataset, without requiring explicit parametrization. We further provide a semidefinite programming formulation to verify consistency with non-degenerate interconnection and dissipation structures. Our results provide a principled approach to assess instant data consistency with physical structure and pave the way for control design directly from data.

I. INTRODUCTION

The paper extends data-driven consistency analysis to nonlinear port-Hamiltonian systems without explicit parametrization. It derives data-only interpolation conditions and a tractable SDP formulation for structured physical models.

Existing nonlinear data-driven methods often target specific system classes or rely on predefined bases, lifting maps, or parametrized function classes.
The framework tests whether observed trajectories are instantly consistent with a structured dynamical system by combining measurements with prior physical insight.
Port-Hamiltonian systems encode energy conservation, dissipation, interconnection structure, and passivity through an energy function and structure matrices.
Interpolation conditions characterize finite data points that can be matched by smooth or smooth convex Hamiltonians without parametrizing the dynamics.
The full-rank case admits an equivalent convex semidefinite programming formulation, enabling tractable consistency verification.

II. PROBLEM SETTING

The problem setting uses continuous-time measurements and a known input matrix to assess noiseless pH consistency. The Hamiltonian is treated nonparametrically through smoothness, convexity, and nonnegativity classes.

The dataset contains state, derivative, and input measurements collected from a continuous-time time-invariant dynamical system.
The analysis asks whether the data could have been generated by a noiseless port-Hamiltonian system with a given constant input matrix.
Nonnegative Hamiltonians support passivity with storage function H, while pH structure also represents energy-preserving interconnections and dissipation.
Rather than using basis expansions or neural approximations, the framework characterizes the Hamiltonian through regularity properties such as smoothness and strong convexity.
The chosen function classes impose gradient-variation bounds and, under strong convexity, structural properties including uniqueness of equilibria and stability robustness.

A. Objectives

The objectives are to characterize instant consistency with smooth and strongly convex pH Hamiltonians, then identify conditions for the non-degenerate full-rank subclass.

Instant consistency requires matrices J and R and a Hamiltonian in the selected nonnegative class to satisfy the pH dynamics at every measured triple.
The paper seeks necessary and sufficient algebraic conditions for consistency with pH systems having L-smooth or µ-strongly convex L-smooth Hamiltonians.
A separate objective is to derive necessary and sufficient conditions when J − R is nonsingular.

1) Structural assumptions:

The structural assumptions fix the input, interconnection, and damping matrices as constant, while separately considering the full-rank condition on J − R.

Structural assumptions:: The input matrix G is assumed known and constant, representing known actuator configuration with unknown internal dynamics.
Structural assumptions:: Interconnection and damping matrices are restricted to constants for tractability while preserving essential geometric structure.
Structural assumptions:: Full rank of J − R is equivalent to ker J ∩ ker R = {0}, so every direction is affected by interconnection, dissipation, or both.
Structural assumptions:: Strictly positive definite damping guarantees invertibility of J − R, and in graph-based pH systems corresponds to acyclic topologies.

2) Noise and derivative estimation:

The framework currently assumes noise-free state and derivative measurements, treating consistency as perfect interpolation. Noisy data and trajectory-level coherence are left for future extensions.

2) Noise and derivative estimation:: The preliminary framework assumes noise-free measurements of both the state and its derivative.This corresponds to a perfect interpolation setting.
2) Noise and derivative estimation:: When derivatives are unavailable, filtering or numerical differentiation can estimate them, while discrete-time formulations preserve the continuous-time structure.Derivative estimation introduces noise; discrete-time formulations provide an alternative formulation.
2) Noise and derivative estimation:: Instant consistency treats state and derivative samples independently, so it is necessary but insufficient for trajectory consistency.It does not enforce the relationship ˙x = dx/dt.

3) Instant consistency:

Instant consistency can produce false positives because it does not preserve temporal ordering, but derivative estimation and sufficiently dense true-derivative data reduce this risk.

3) Instant consistency:: Temporal-order invariance allows a non-pH system to appear consistent when its samples admit a suitable permutation.This is a concrete source of false positives in pointwise consistency checking.
3) Instant consistency:: Numerical differentiation implicitly enforces state-derivative relationships but introduces noise, whereas discrete-time formulations enforce exact temporal coupling under their assumptions.Discrete-time conditions remain exact, although discretization violations can prevent the resulting model from matching the true system.
3) Instant consistency:: For sufficiently dense data with true measured derivatives, the discrepancy between instant and trajectory consistency is expected to vanish except in highly degenerate cases.The framework therefore remains useful for ruling out model classes fundamentally incompatible with observed physics.
3) Instant consistency:: Enforcing trajectory-level coherence in continuous time remains a fundamental direction for future research.The current framework does not explicitly impose this coherence.

C. Interpolation conditions

Interpolation conditions characterize whether finite data admit a smooth or smooth convex Hamiltonian without explicitly constructing the function. The resulting criteria are necessary and sufficient.

C. Interpolation conditions: Interpolation conditions determine whether finite data are consistent with a prescribed function class.They guarantee the existence of a function in that class interpolating the supplied data points.
C. Interpolation conditions: The interpolation dataset consists of state points, gradient values, and function values represented as triples (x_i, g_i, h_i).A function H interpolates the set when g_i = ∇H(x_i) and H(x_i) = h_i for every index.
C. Interpolation conditions: The adapted interpolation conditions characterize when a function with the required smoothness and convexity properties exists.Proposition 1 gives conditions for smooth and smooth convex interpolation.
C. Interpolation conditions: These conditions are necessary and sufficient for observed data to be instantly consistent with a Hamiltonian having specified smoothness and convexity properties.Thus, consistency can be checked from algebraic relations among the data points.

III. MAIN RESULTS

The main results characterize instant consistency of measured data with smooth or convex port-Hamiltonian systems through necessary and sufficient algebraic conditions. They also provide an SDP formulation for testing consistency with full-rank pH systems and recovering a compatible realization.

General consistency conditions: The matrix A = J − R represents interconnection and dissipation when A + A⊤ ⪯ 0.This single negative-semidefinite symmetric-part condition is equivalent to the existence of skew-symmetric J and positive-semidefinite R.
General consistency conditions: Theorem 1 reduces instant consistency with smooth or strongly convex pH systems to feasibility of data-dependent algebraic conditions.The conditions combine a structured operator representation with interpolation constraints for the Hamiltonian.
Convex Hamiltonians: For convex L-smooth Hamiltonians, consistency is equivalent to existence of A, gradients gi, and energy values hi satisfying the data equation and interpolation inequalities.The convex case uses the condition hi − hj ≥ gj⊤(xi − xj) + 1/(2L)||gi − gj||^2 for i ≠ j.
Computational formulation: The feasibility characterization is generally a nonconvex QCQP because Y = AΓ is bilinear in the unknown operator and candidate gradients.Known parametric interconnection and dissipation structures can be added as constraints; linear parametrizations yield a convex QCQP.
Full-rank systems and SDP: Full-rank pH consistency admits a convex SDP reformulation by imposing A + A⊤ ≺ 0 and substituting gradients gi = Byi with B = A^-1.The resulting quadratic constraints in B are converted to LMIs through the Schur complement, and a compatible realization can be recovered when the SDP is feasible.
Full-rank systems and SDP: Algorithm 1 computes yi and pairwise differences, solves the SDP, and returns feasibility together with A, gradients, and energy values when available.The framework can also relax perfect interpolation with slack variables to account for noisy measurements.

IV. NUMERICAL SIMULATIONS

The simulations evaluate Algorithm 1 on convex and non-convex nonlinear pH systems under varying smoothness, derivative estimation, sampling, and data-density conditions. The framework identifies the convex pH benchmark and rejects convexity for the non-convex actuator, while relaxed smoothness and coarse sampling can produce false feasibility.

Experimental setup: Algorithm 1 is evaluated on a mass-spring-damper system with a convex Hamiltonian and an electrostatic microactuator with a non-convex Hamiltonian.The experiments assess structure identification, parameter sensitivity, derivative estimation, instant consistency, and sampling effects.
Experimental setup: The experiments use sinusoidal input, 100 recorded data pairs, multiple sample counts and spacings, and three derivative-estimation methods.The tested derivatives are exact, central finite differences with O(∆t^2), and structure-preserving midpoint discretization.
Results: The convex benchmark remains feasible for µ-strongly convex Hamiltonians at L = 4 when µ = 0 and L = 6 when µ = 0.1.These are reported as relatively tight smoothness bounds for Example 4.1.
Results: The non-convex actuator is rejected as µ-strongly convex for small L, but overly relaxed smoothness constraints create false feasibility.Figure 2 presents the corresponding feasibility regions for the non-convex electrostatic microactuator.
Results: Midpoint discretization most closely matches the true feasibility boundary, whereas central finite differences can introduce false feasibility at larger L.All derivative-estimation methods show similar overall behavior, but discretization error affects the boundary comparison.
Results: The number of sampled points has negligible impact for the non-convex example but relevant impact for the convex example, while larger dt reduces accuracy.Coarser sampling is associated with the reported false-convexity behavior.

V. DISCUSSION: TOWARDS DATA-DRIVEN CONTROL VIA INTERPOLATION

The interpolation viewpoint extends from instant consistency toward data-driven passivity-based control by embedding desired energy shaping and closed-loop structure into data constraints. This direction avoids explicit model parametrization but remains limited by more complex constraints, pointwise enforcement, and the lack of guaranteed stabilizability.

Control extension: The proposed extension targets energy shaping through IDA-PBC, reshaping the energy function and system structure around a desired equilibrium.The desired closed-loop Hamiltonian is required to have a minimum at the prescribed equilibrium with suitable regularity.
Control extension: Control objectives can be embedded directly into interpolation constraints on gradients and energy values consistent with the desired closed-loop structure.The constraints enforce that input-unaffected dynamics components match the desired energy-shaped structure.
Advantages: The viewpoint bypasses explicit model identification or parametrization by replacing it with feasibility conditions checked directly from data.This preserves the data-driven consistency perspective while incorporating physical structure.
Advantages: Interpolating gradients and energy values may support controller construction consistent with both measured data and desired equilibrium behavior.This is presented as a possibility rather than a completed control-synthesis result.
Challenges: Closed-loop constraints are more complex than open-loop constraints because they involve bilinear A(d)g(d) terms and additional input-direction degrees of freedom.These structural complications are identified as challenges for the proposed control extension.
Challenges: Interpolation conditions enforce consistency only at sampled points and do not guarantee desirable behavior between samples or stabilizability.Compatibility with IDA-PBC at the data level is described as necessary but not sufficient for stabilizability.

VI. CONCLUSIONS

The paper develops interpolation-based tests for instant pH consistency directly from measured trajectories and gives necessary and sufficient algebraic conditions, including a convex SDP formulation in the full-rank case. It also identifies pointwise enforcement and the absence of trajectory-level coherence as limitations while motivating extensions to data-driven passivity-based control.

Conclusions: The framework assesses instant consistency with pH systems directly from measured trajectories without explicit model identification.It uses interpolation conditions for smooth and convex function classes.
Conclusions: Necessary and sufficient algebraic conditions characterize consistency with structured dynamical models.The conditions are derived using interpolation conditions for smooth and convex Hamiltonians.
Conclusions: The full-rank case admits a convex semidefinite programming formulation for efficient verification from data.This formulation addresses non-degenerate pH structure in the reported setting.
Future directions: The framework may extend toward data-driven control, particularly through passivity-based control approaches.The conclusion presents this as an extension beyond the current consistency analysis.
Limitations: The current framework is pointwise and does not enforce trajectory-level coherence; noisy and trajectory-consistent extensions remain future work.This is identified as an important limitation and direction for further development.